The structure of almost-invariant half-spaces for some operators

摘要

Given a Banach space Xand an operator T∈L(X), an almost-invariant half-space (AIHS) is a closed subspace Ysatisfying TY?Y+F for some finite-dimensional subspace Fand dim(Y) =dim (X/Y) =∞. In this paper we study the connection between AIHS and noncompactness of some subsets of the unit ball, chains of T-invariant subspaces, and different parts of spectrum of Tand T?. We describe a simple template for such subspaces and show that if Tis quasinilpotent or weakly compact then it admits an AIHS Y.